There is a natural analogue of the codimension–1 Riesz kernel in the Heisenberg group \(\mathbb {H}\) . Recall that in \(\mathbb {R}^n\)  the Riesz kernel \(R_{n-1}(x):=x|x|^{-n}\)  is a constant multiple of the gradient of the fundamental solution of the Laplacian.
Sub-Riemannian analogues of the Laplacian, known as sub-Laplacians, have been extensively studied in Carnot groups and sub-Riemannian manifolds since the early 70s and the works of Stein, Folland, and others [1]}, [2]}, [2]}. A thorough treatment of this fully-fledged theory can be found in [4]}. In particular, the (canonical) sub-Laplacian in \(\mathbb {H}\)  is defined as
\(\Delta _{\mathbb {H}}=X_\mathsf {L}^2+Y_\mathsf {L}^2,\) 
